On the intersection of maximalm-convex subsets

A subsetS of a real linear spaceE is said to bem-convex providedm≧2, there exist more thanm points inS, and for eachm distinct points ofS at least one of the ( ₂ ^m ) segments between thesem points is included inS. InE, letxy denote the segment between two pointsx andy. For any pointx inSυE, letS _x ={y: xyυS}. The kernel of a setS is then defined as {xεS: S _x=S}. It is shown that the kernel of a setS is always a subset of the intersection of all maximalm-convex subsets ofS. A sufficient condition is given for the intersection of all the maximalm-convex subsets of a setS to be the kernel ofS.